Abstract
Despite their successes, machine learning techniques are often stochastic, error-prone and blackbox. How could they then be used in fields such as theoretical physics and pure mathematics for which error-free results and deep understanding are a must? In this Perspective, we discuss techniques for obtaining zero-error results with machine learning, with a focus on theoretical physics and pure mathematics. Non-rigorous methods can enable rigorous results via conjecture generation or verification by reinforcement learning. We survey applications of these techniques-for-rigor ranging from string theory to the smooth 4D Poincaré conjecture in low-dimensional topology. We also discuss connections between machine learning theory and mathematics or theoretical physics such as a new approach to field theory motivated by neural network theory, and a theory of Riemannian metric flows induced by neural network gradient descent, which encompasses Perelman’s formulation of the Ricci flow that was used to solve the 3D Poincaré conjecture.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$99.00 per year
only $8.25 per issue
Buy this article
- Purchase on Springer Link
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
References
Jumper, J. et al. Highly accurate protein structure prediction with AlphaFold. Nature 596, 583–589 (2021).
Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).
Ruehle, F. Data science applications to string theory. Phys. Rept. 839, 1–117 (2020).
He, Y. Machine Learning in Pure Mathematics and Theoretical Physics (World Scientific, 2023).
Athalye, A., Engstrom, L., Ilyas, A. & Kwok, K. Synthesizing robust adversarial examples. Proc. Mach. Learn. Res. 80, 284–293 (2018).
Athalye, A., Carlini, N. & Wagner, D. Obfuscated gradients give a false sense of security: circumventing defenses to adversarial examples. Preprint at https://arxiv.org/abs/1802.00420 (2018).
Gukov, S., Halverson, J., Manolescu, C. & Ruehle, F. Searching for ribbons with machine learning. Preprint at https://doi.org/10.48550/arXiv.2304.09304 (2023).
Neal, R. M. Bayesian Learning for Neural Networks. PhD thesis, Univ. Toronto (1995).
Jacot, A., Gabriel, F. & Hongler, C. Neural tangent kernel: convergence and generalization in neural networks. In 32nd Conference on Neural Information Processing Systems (eds Garnett, R. et al.) 1–10 (2018).
Lee, J. et al. Wide neural networks of any depth evolve as linear models under gradient descent. J. Statist. Mech. 2020, 124002 (2019).
Demirtas, M., Halverson, J., Maiti, A., Schwartz, M. D. & Stoner, K. Neural network field theories: non-Gaussianity, actions, and locality. Mach. Learn. Sci. Technol. 5, 015002 (2023).
Perelman, G. The entropy formula for the Ricci flow and its geometric applications. Preprint at https://doi.org/10.48550/arXiv.math/0211159 (2002).
Hamilton, R. S. Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982).
Gauthier, T., Kaliszyk, C., Urban, J., Kumar, R. & Norrish, M. TacticToe: learning to prove with tactics. J. Automat. Reason. 65, 257–286 (2018).
Szegedy, C. (ed.). A Promising Path Towards Autoformalization and General Artificial Intelligence (Springer, 2020).
Wu, Y. et al. Autoformalization with large language models. Preprint at https://doi.org/10.48550/arXiv.2205.12615 (2022).
Lample, G. et al. HyperTree proof search for neural theorem proving. Preprint at https://doi.org/10.48550/arXiv.2205.11491 (2022).
Hales, T. C. Developments in formal proofs. Asterisque Exp. No. 1086, 387–410 (2015).
Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E. & Urban, J. Premise selection for mathematics by corpus analysis and kernel methods. J. Automat. Reason. 52, 191–213 (2014).
Blanchette, J. C., Greenaway, D., Kaliszyk, C., Kühlwein, D. & Urban, J. A learning-based fact selector for Isabelle/HOL. J. Automat. Reason. 57, 219–244 (2016).
Nagashima, Y. Simple dataset for proof method recommendation in Isabelle/HOL (dataset description). In Intelligent Computer Mathematics: 13th International Conference 297–302 (ACM, 2020).
Piotrowski, B., Mir, R. F. & Ayers, E. Machine-learned premise selection for lean. Preprint at https://doi.org/10.48550/arXiv.2304.00994 (2023).
Carifio, J., Halverson, J., Krioukov, D. & Nelson, B. D. Machine learning in the string landscape. J. High Energy Phys. 09, 157 (2017).
He, Y.-H. Deep-learning the landscape. Phys. Lett. B 774, 564–568 (2017).
Krefl, D. & Seong, R.-K. Machine learning of Calabi-Yau volumes. Phys. Rev. D 96, 066014 (2017).
Ruehle, F. Evolving neural networks with genetic algorithms to study the string landscape. J. High Energy Phys. 08, 038 (2017).
Davies, A. et al. Advancing mathematics by guiding human intuition with ai. Nature 600, 70–74 (2021).
Craven, J., Jejjala, V. & Kar, A. Disentangling a deep learned volume formula. J. High Energy Phys. 06, 040 (2021).
Craven, J., Hughes, M., Jejjala, V. & Kar, A. Learning knot invariants across dimensions. SciPost Phys. 14, 021 (2023).
Brown, G. et al. Computation and data in the classification of Fano varieties. Preprint at https://doi.org/10.48550/arXiv.2211.10069 (2022).
Mishra, C., Moulik, S. R. & Sarkar, R. Mathematical conjecture generation using machine intelligence. Preprint at https://doi.org/10.48550/arXiv.2306.07277 (2023).
Cranmer, M. D. et al. Discovering symbolic models from deep learning with inductive biases. In Advances in Neural Information Processing Systems 33 (NeurIPS, 2020).
Silver, D. et al. Mastering the game of Go without human knowledge. Nature 550, 354–359 (2017).
Silver, D. et al. Mastering chess and shogi by self-play with a general reinforcement learning algorithm (2017). Preprint at https://doi.org/10.48550/arXiv.1712.01815 (2017).
Strogatz, S. One giant step for a chess-playing machine. The New York Times https://www.nytimes.com/2018/12/26/science/chess-artificial-intelligence.html (2018).
Klaewer, D. & Schlechter, L. Machine learning line bundle cohomologies of hypersurfaces in toric varieties. Phys. Lett. B 789, 438–443 (2019).
Brodie, C. R., Constantin, A., Deen, R. & Lukas, A. Topological formulae for the zeroth cohomology of line bundles on del Pezzo and Hirzebruch surfaces. Compl. Manif. 8, 223–229 (2021).
Brodie, C. R., Constantin, A., Deen, R. & Lukas, A. Index formulae for line bundle cohomology on complex surfaces. Fortsch. Phys. 68, 1900086 (2020).
Brodie, C. R., Constantin, A., Deen, R. & Lukas, A. Machine learning line bundle cohomology. Fortsch. Phys. 68, 1900087 (2020).
Brodie, C. R. & Constantin, A. Cohomology chambers on complex surfaces and elliptically fibered Calabi-Yau three-folds. Preprint at https://doi.org/10.48550/arXiv.2009.01275 (2020).
Bies, M. et al. Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory. J. High Energy Phys. 01, 196 (2021).
Halverson, J., Nelson, B. & Ruehle, F. Branes with brains: exploring string vacua with deep reinforcement learning. J. High Energy Phys. 06, 003 (2019).
Cole, A., Krippendorf, S., Schachner, A. & Shiu, G. Probing the structure of string theory vacua with genetic algorithms and reinforcement learning. In 35th Conference on Neural Information Processing Systems (NeurIPS, 2021).
Krippendorf, S., Kroepsch, R. & Syvaeri, M. Revealing systematics in phenomenologically viable flux vacua with reinforcement learning. Preprint at https://doi.org/10.48550/arXiv.2107.04039 (2021).
Abel, S., Constantin, A., Harvey, T. R. & Lukas, A. String model building, reinforcement learning and genetic algorithms. In Nankai Symposium on Mathematical Dialogues (iNSPIRE, 2021).
Abel, S., Constantin, A., Harvey, T. R. & Lukas, A. Evolving heterotic gauge backgrounds: genetic algorithms versus reinforcement learning. Fortsch. Phys. 70, 2200034 (2022).
Constantin, A., Harvey, T. R. & Lukas, A. Heterotic string model building with monad bundles and reinforcement learning. Fortsch. Phys. 70, 2100186 (2022).
Hughes, M. C. A neural network approach to predicting and computing knot invariants. J. Knot Theory Ramif. 29, 2050005 (2020).
Sundararajan, M., Taly, A. & Yan, Q. Axiomatic attribution for deep networks. In Proc. 34th International Conference on Machine Learning (eds Precup, D. & Teh, Y. W.) Vol. 70, 3319–3328 (PMLR, 2017).
Hass, J., Lagarias, J. C. & Pippenger, N. The computational complexity of knot and link problems. J. ACM 46, 185–211 (1999).
Kuperberg, G. Knottedness is in NP, modulo GRH. Adv. Math. 256, 493–506 (2014).
Lackenby, M. The efficient certification of knottedness and Thurston norm. Adv. Math. 387, 107796 (2021).
Gukov, S., Halverson, J., Ruehle, F. & Sułkowski, P. Learning to unknot. Mach. Learn. Sci. Technol. 2, 025035 (2021).
Alexander, J. W. A lemma on systems of knotted curves. Proc. Natl Acad. Sci. USA 9, 93–95 (1923).
Ri, S. J. & Putrov, P. Graph neural networks and 3-dimensional topology. Mach. Learn. Sci. Tech. 4, 035026 (2023).
Gukov, S., Halverson, J., Manolescu, C. & Ruehle, F. An algorithm for finding ribbon bands. GitHub https://github.com/ruehlef/ribbon (2023).
Williams, C. K. In Advances in Neural Information Processing Systems 295–301 (1997).
Yang, G. Tensor programs I: wide feedforward or recurrent neural networks of any architecture are Gaussian processes. In Advances in Neural Information Processing Systems 32 (NeurIPS, 2019).
Roberts, D. A., Yaida, S. & Hanin, B. The Principles of Deep Learning Theory: an Effective Theory Approach to Understanding Neural Networks (Cambridge Univ. Press, 2022).
Halverson, J., Maiti, A. & Stoner, K. Neural networks and quantum field theory. Mach. Learn. Sci. Tech. 2, 035002 (2021).
Halverson, J. Building quantum field theories out of neurons. Preprint at https://doi.org/10.48550/arXiv.2112.04527 (2021).
Osterwalder, K. & Schrader, R. Axioms for Euclidean green’s functions. Commun. Math. Phys. 31, 83–112 (1973).
Erbin, H., Lahoche, V. & Samary, D. O. Non-perturbative renormalization for the neural network-QFT correspondence. Mach. Learn. Sci. Tech. 3, 015027 (2022).
Grosvenor, K. T. & Jefferson, R. The edge of chaos: quantum field theory and deep neural networks. SciPost Phys. 12, 081 (2022).
Banta, I., Cai, T., Craig, N. & Zhang, Z. Structures of neural network effective theories. Preprint at https://doi.org/10.48550/arXiv.2305.02334 (2023).
Krippendorf, S. & Spannowsky, M. A duality connecting neural network and cosmological dynamics. Mach. Learn. Sci. Technol. 3, 035011 (2022).
Maiti, A., Stoner, K. & Halverson, J. in Machine Learning in Pure Mathematics and Theoretical Physics Ch. 8, 293–330 (2023).
Halverson, J. & Ruehle, F. Metric flows with neural networks. Preprint at https://doi.org/10.48550/arXiv.2310.19870 (2023).
Anderson, L. B. et al. Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning. J. High Energy Phys. 05, 013 (2021).
Douglas, M. R., Lakshminarasimhan, S. & Qi, Y. Numerical Calabi-Yau metrics from holomorphic networks. In Proc. 2nd Mathematical and Scientific Machine Learning Conference Vol. 145, 223–252 (PMLR, 2022).
Jejjala, V., Mayorga Pena, D. K. & Mishra, C. Neural network approximations for Calabi-Yau metrics. J. High Energy Phys. 08, 105 (2022).
Larfors, M., Lukas, A., Ruehle, F. & Schneider, R. Learning size and shape of Calabi-Yau spaces. In Fourth Workshop on Machine Learning and the Physical Sciences (2021).
Larfors, M., Lukas, A., Ruehle, F. & Schneider, R. Numerical metrics for complete intersection and Kreuzer–Skarke Calabi–Yau manifolds. Mach. Learn. Sci. Tech. 3, 035014 (2022).
Gerdes, M. & Krippendorf, S. CYJAX: a package for Calabi-Yau metrics with JAX. Mach. Learn. Sci. Tech. 4, 025031 (2023).
Yau, S.-T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation, I. Commun. Pure Appl. Math. 31, 339–411 (1978).
Calabi, E. On Kähler Manifolds with Vanishing Canonical Class 78–89 (Princeton Univ. Press, 2015).
Donaldson, S. K. Some numerical results in complex differential geometry (2005). Preprint at https://doi.org/10.48550/arXiv.math/0512625 (2005).
Yang, G. Tensor programs II: neural tangent kernel for any architecture. Preprint at https://doi.org/10.48550/arXiv.2006.14548 (2020).
Cotler, J. & Rezchikov, S. Renormalization group flow as optimal transport. Phys. Rev. D https://doi.org/10.1103/physrevd.108.025003 (2023).
Berman, D. S. & Klinger, M. S. The inverse of exact renormalization group flows as statistical inference. Preprint at https://doi.org/10.48550/arXiv.2212.11379 (2022).
Berman, D. S., Klinger, M. S. & Stapleton, A. G. Bayesian renormalization. Mach. Learn. Sci. Technol. 4, 045011 (2023).
Acknowledgements
S.G. is supported in part by a Simons Collaboration Grant on New Structures in Low-Dimensional Topology and by the Department of Energy grant DE-SC0011632. J.H. is supported by National Science Foundation CAREER grant PHY-1848089. F.R. is supported by the National Science Foundation grants PHY-2210333 and startup funding from Northeastern University. The work of J.H. and F.R. is also supported by the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions).
Author information
Authors and Affiliations
Contributions
The authors contributed equally to all aspects of the article.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Reviews Physics thanks Marika Taylor, Sven Krippendorf and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gukov, S., Halverson, J. & Ruehle, F. Rigor with machine learning from field theory to the Poincaré conjecture. Nat Rev Phys 6, 310–319 (2024). https://doi.org/10.1038/s42254-024-00709-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s42254-024-00709-0